Even though geographically it does not seem obvious, my favorite, London City Airport (LCY) is also easily accessible if you use public transport (about 1h), and the very short boarding/deplaning probably make the total time between plane and Harrow the shortest. But there are less destinations served from LCY.
– jcaronDec 13 '18 at 9:12

@jcaron Unfortunately LCY is only served by very few airlines from very few destination.
– Aleks GDec 13 '18 at 16:53

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I removed the part about commuting to Harrow on the HIll, since that's a completely separate question, and it's not been addressed in any of the answers. If you want, you can ask it as a separate question. However, If you ask Google Maps how to get from Harrow to Harrow on the Hill, it says it's only about 1.5km. At that point, the best way to get between them will depend very much on where in Harrow you're starting from and where in Harrow on the Hill you're going to. The answer might be "walk for five minutes" or "take a bus from the other side of Harrow." Without more precise detail, ...
– David RicherbyDec 13 '18 at 22:16

4 Answers
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There are 5 main airports in London. Gatwick is SWish, London City is in the inner east, Luton is north, Stansted is north east and far. Heathrow is in the west, and about an hour away from Harrow by public transport.

For fun, I plugged them all into Google to check times by transport (public).

Gatwick is 25 miles from central London; Luton is 27.5 and Stansted is 30, which isn't a whole lot farther. Of the three, Gatwick has by far the best transport links to central London but it's not really accurate to describe only Stansted as "far".
– David RicherbyDec 13 '18 at 15:13

Geographically of the major London airports Heathrow seems to be closest. If you plan to take a cab or rent a car then Heathrow is probably the best, followed by Stansted and Luton. If you want to travel by car I would avoid Gatwick (other side of London) and London City (as the name suggests in the city centre) airports.

On the other hand if you plan to travel by public transport then distance often bears little relationship to travel time. Busses are slow and trains tend to go to/from central London. Neither of the Harrow rail stations is on the same rail line as any of the airports. Heathrow still seems to be the best, but after that as Mark points out it's a wash in terms of travel time (though probably not travel cost) between City, Gatwick, Stansted and Luton.

If using a car, Luton is much closer than Stansted, which in turn is about the same as Gatwick from Harrow. Though of course this probably depends a lot on time, day of the week, and direction.
– jcaronDec 13 '18 at 12:54

I don't think you can reasonably describe London City as being "in the city centre": it's only just inside the North Circular.
– David RicherbyDec 13 '18 at 15:21

As Mark Mayo's answer shows, except for Stansted, the travel times from the major London airports to Harrow are basically the same. The differences between them aren't significant compared to other aspects of your journey. For example, flying at a time that's more convenient for you, picking a cheaper flight or a route with better connections would probably all make a bigger difference than the possibility of saving 15 minutes travelling from the airport to Harrow.

Mark Mayo provide a good factual answer about London's airports and David Richerby added a couple of good subjective points (cheaper flights, time convenience, etc.)

I would add that, from personal experience, if you are planning to travel by car, then Luton or Heathrow would be best, as they are roughly the same distance from Harrow and you'd be travelling through roughly the same volumes of traffic (possibly, a bit less traffic from Luton).

If you are planning on using public transport, then I would recommend Heathrow, as it's much better connected, with a variety of public transport options and you can get from Heathrow to Harrow on the Hill using only tube and public bus. Have a look at google map options and do note that on the Piccadilly line you can switch at Acton Town station, no need to go to Hammersmith - this would save you another 10 minutes or so.